CaptainBlack what you said is true. I made a mistake in how I posed my question, I wanted to say a subgroup not a subset.

The revised question:
1)Let be a group.
2)Let .
3)Let .
4)Then is a topology on .
The reason is because the union/intersection of two subgroups is a subgroup.

You had better tell us what you want the expressions "union of two subgroups"
and "intersection of two subgroups" to mean. As it is they don't mean
anything as union and intersection are usually defined on sets. And a
groups is not a set, a set may be involved but the group is not the set,
and as rgep points out the naive union/intersection of the associated sets
of a pair of subgroups is not usually the set associated with another subgroup.

Maybe I did make a mistake. I just remember my book on Abstract Algebra asking to prove the intersection/uninon of groups, is a group something like that. Maybe, it is true for Normal Subgroups I do not remember exactly.

I do not think I made a mistake. By uninon/intersection I mean respectively. I just am visualizing that in my head, I think it is a subgroup.

Maybe I did make a mistake. I just remember my book on Abstract Algebra asking to prove the intersection/uninon of groups, is a group something like that. Maybe, it is true for Normal Subgroups I do not remember exactly.

I do not think I made a mistake. By uninon/intersection I mean respectively. I just am visualizing that in my head, I think it is a subgroup.

But is not in general closed under the group operation, and
so in general not a subgroup.

You might want to consider the group with the following group multiplication
table (which if I have done this right makes a group):

Then: , , , are all subgroups of ,
but is not a group because
is not closed under